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Full text of "Analytical Mechanics"

MOTION 97 f = limit Vr* S*n €2 "*" ^ COS €2 ~ ^ """ VTI s*n €1 "^Pl CQS 6l^ p *=o = limit dr . d , N = <°^+^(no) -i|(r.«), (XV) where co is the angular velocity of the radius vector. PROBLEMS. 1. A particle describes the parabola y = 2 px so that its velocity along the x-axis is constant and equals u. Find the total velocity and the acceleration. 2. Discuss the motions defined by the following equations deriving the expressions for the path, velocity, acceleration, and the various components of the last two : (a) x = at, y = b v7. (d) x = at, y = be~kt. (b) x = at, y = U — | gtz. (e) x = at, y = b sin cot. (c) x = aekt, y = frefc*. (f) x = acosco£, ?/ = Ztf- 3. Express in terms of t the velocity and the acceleration of a particle which moves so that x = ay and y = ax. 4. A particle describes a circle of radius a with a constant speed v. Find /,,/y,/.,/p,/r,/n,/r, and/. 6. Work out the preceding problem graphically. 91. Angular Acceleration. — Angular acceleration is the time rate at which angular velocity changes. Therefore, denoting it by T, we have dco T= — = co !> , (xvi) dt2 ' . If the angular velocity of a body increases uniformly in one second the body is said to have a unit angular .1 sec.